Introduction to Astronomy

Lecture 5: The Motion of the Planets

The heavens themselves, the planets, and this centre
Observe degree, priority, and place,
Insisture, course, proportion, season, form,
Office, and custom, in all line of order:

-- William Shakespeare, Troilus and Cressida, 1609

5.1 Direct and Retrograde Motion

(Discovering the Universe, 5th ed., §2-0)

In addition to the stars, the Sun, and the Moon, there are several other objects in the sky which are easily visible at night.
From the ancient perspective, a planet is a point of light in the sky that moves relative to the stars, much as the Sun and Moon do.

5.2 Geocentric Cosmology

(Discovering the Universe, 5th ed., §2-0)

According to the laws of physics, there is no preference
between saying that the Sun revolves around the Earth or the
Earth revolves around the Sun.

Each are equally true, although one perspective may be more useful
than the other for a particular purpose, as we have seen.

When it comes to the planets, however, the principles of
science forces one to make a distinction between an Earth-based
view and a Sun-based view.

Given that the motion of the Earth cannot be perceived, it
is natural to assume the planets revolve around it. This is known
as a geocentric cosmology, and it was widely accepted
until just a few hundred years ago.

A good scientist, however, must try and make sense out of
retrograde motion, which cannot be simply explained in an Earth-centered
perspective.

The
Greek astronomer Hipparchus
described a model for retrograde motion which placed each planet
in motion around a circle called an epicycle, which in
turn revolved around a circle centered on the Earth called a
deferent.

The former produces the retrograde motion, while the latter is
primarily responsible for the direct motion.

In the adjacent animation, the arrow points towards the stars
we see behind the planet; watch where the arrow points as the
planet moves around its epicycle.

In the 1st C. A.D., Ptolemy,
an astronomer at the Alexandria observatory in Egypt, took Hipparchus'
model and fit it to the several centuries of observational data
that was available to him.

For each planet, Ptolemy determined the sizes of its deferent
and epicycle, and the speed of revolution of its epicycle and
the planet itself.

By projecting his model forward in time, Ptolemy was then able
to correctly predict where the planets would be located centuries
into the future.

Because of the success of his model, Ptolemy's treatise on the
subject, which became known as the Almagest ("the
Greatest"), was the bible of astronomers through the Middle
Ages.

After a millenium, however, the Ptolemaic model increasingly
deviated from the planets' observed motions.

Other astronomers tried to correct it by adding additional levels
of epicycles, but the result was exceedingly complex.

It was clear that something was not quite right with this model.

5.3 Heliocentric Cosmology

(Discovering the Universe, 5th ed., §2-1)

An alternative to the geocentric cosmology was actually suggested
a century before Hipparchus by Aristarchus
(3rd C. B.C.).

In the heliocentric cosmology, the planets orbit the
Sun rather than the Earth.

The Earth also orbits the Sun, but the Moon still orbits the
Earth.
Mercury and Venus (the inner planets) have smaller orbits
than the Earth, while Mars, Jupiter, and Saturn (the outer
planets) have larger orbits.

The
heliocentric cosmology also explains retrograde motion, by relying
on the fact that the planets move at different speeds; in particular,
inner planets move faster and outer planets move more slowly.

As a result, the Earth will regularly overtake and pass the outer
planets, and the inner planets will do the same to the Earth.

Like one car passing another on the highway, the second car will
appear to "move backward".

Although the heliocentric cosmology had a simpler geometry
than the geocentric cosmology, it never achieved acceptance in
ancient Greece because it required that the Earth move.

The heliocentric cosmology was forgotten for almost 2000 years,
until the 16th century, when the Polish astronomer Nicolaus
Copernicus rediscovered it.

Copernicus performed his own calculations using this model
(assuming circular orbits), and found that he could describe
the planets' observed motions to an accuracy similar to that
of the geocentric model.

He was also able to make very accurate predictions of the planets'
relative distances from Sun (see the table
below).

Although Copernicus was convinced that his heliocentric model
was a well-founded improvement to astronomy, he delayed publication
of his results until near the end of his life, presumably because
he was concerned that his work was so radical that it would be
rejected or lead to censure.

Finally, in 1543, Copernicus' book On the Revolutions of the
Celestial Spheres appeared, shortly before he died.

The book was widely read in Europe, and gained enough support
that it seriously threatened the geocentric model, which was
virtually an article of faith in the Catholic Church.

In 1616 the Church banned Copernicus' book (and the ban was not
lifted until the end of the 18th century!).

5.4 Planetary Configurations

(Discovering the Universe, 5th ed., §2-1)

For millenia, observational astronomers have described the
positions of planets and other celestial bodies using several
special configurations, which can be easily understood in terms
of the Copernican model.

The configuration between two celestial bodies can be described
using the angle between them (measured along the ecliptic), which
is called their elongation.

Question: what other type
of celestial angular measurement is elongation similar to?

When two or more celestial bodies pass each other along the
ecliptic, they have an elongation of 0°, and they are said
to be in conjunction.

At the right, the Sun and the red planet are in conjunction.

Question: what is the phase
of the Moon when it is in conjunction with the Sun?

The image at the right shows a "triple" conjunction that occurred on April 23, 1998, between the Moon,Venus, and Jupiter.

When two celestial bodies are at right angles in the sky, they have an elongation of 90°, and they are said to be in quadrature.

At the right, the Sun near the western horizon and the red planet
near the meridian are in quadrature.

Question: what is the phase
of the Moon when it is in quadrature with the Sun?

When two celestial bodies are directly opposite each other,
they have an elongation of 180°, and they are said to be
in opposition.

At the right, the Sun near the western horizon and the red planet
near the eastern horizon are in opposition.

Question: what is the phase
of the Moon when it is in opposition to the Sun?

Mercury and Venus differ from the other planets in that they
never appear far from the Sun.
Mercury is at most 23° away from the Sun, while Venus is
at most 46° away; this is their maximum elongation.

Maximum elongation of a planet is not readily explained using
the geocentric model, but it arises naturally out of the heliocentric
model, simply by assuming that the orbits of Mercury and Venus
lie inside the Earth's orbit.

Maximum elongation of
an inner planet from the Sun can be seen from the geometry of
the picture at the right; it is determined by the tangent line
from the Earth to the planet's orbit.
From the rotation of the Earth we can determine the relative
directions of the planet and the Sun, allowing us to distinguish
the two sides of the planet's orbit as eastern and western.

When an inner planet is at maximum eastern elongation it will
only be visible shortly after sunset (an "evening star");
when it is at maximum western elongation it will only be visible
shortly before sunrise (a "morning star").

Note that an inner planet can never be in opposition to (180°
away from) the Sun, or even in quadrature (90° away).

For the planet and the Sun to be in conjunction, they must
be along the same line of sight from the Earth.
From the picture above we can see that there are two different
ways in which an inner planet can be in conjunction with the
Sun, one in between the Sun and the Earth (called inferior
conjunction) and the other on the opposite side of the Sun
from the Earth (called superior conjunction).

Question: as observed from
Earth, what is the phase
of the inner planet at each of the four positions in the picture?

Unlike Mercury and Venus,
Mars, Jupiter, and Saturn can appear in opposition to the Sun
(180° away).

In the heliocentric model, their orbits must therefore lie outside
the Earth's orbit.

An outer planet can be in opposition with the Sun, in quadrature
with it (both eastern and western quadrature),
and in conjunction with it (but only one way, corresponding to
an inner planet's superior conjunction).

Question: as observed from
Earth, what is the phase
of the outer planet at each of the four positions in the picture?

It can be shown that an outer planet is least illuminated at
quadrature.

Question: when will an outer planet be brightest?

5.5 Orbital Period

(Discovering the Universe, 5th ed., §2-1)

The time it takes for a planet to complete one orbit is called
the orbital period of revolution, often simplified to
"orbital period" or just "period".

As with the Moon,
we must distinguish between star-relative and sun-relative positions
when determining the period:
The sidereal period of a planet refers to the time it
takes for the planet to return to the same position with respect
to the stars, e.g. from one position on its orbit back to the
same position.
The synodic period of a planet refers to the time it takes
for the planet to return to the same position with respect to
the Sun, e.g. from inferior conjunction to inferior conjunction,
or from opposition to opposition.

As can be seen in the table at the right, the farther a planet
is from the Sun, the longer is its sidereal period.

The synodic period doesn't have a simple behavior, however; it
initially increases, and then decreases until it is slightly
larger than one year.

This complicated behavior is due to the relative motion of both
the planet and the Earth.

Planet

Average
Distance

Sidereal
Period

Synodic
Period

Mercury

0.3871 AU

0.2408 y = 87.97 d

115.88 d

Venus

0.7233 AU

0.6152 y = 224.70 d

583.92 d

Earth

1.0000 AU

1.0000 y = 365.26 d

----

Mars

1.5237 AU

1.8809 y = 686.98 d

779.94 d

Jupiter

5.2028 AU

11.862 y

398.9 d

Saturn

9.5388 AU

29.458 y

378.1 d

Uranus

19.1914 AU

84.01 y

369.7 d

Neptune

30.0611 AU

164.79 y

367.5 d

Pluto

39.5294 AU

248.5 y

366.7 d

For an inner planet, the sidereal period is shorter than
the synodic period, because when the planet returns to its original
position the Earth has moved in its orbit, so the planet
must travel further to catch up to the Earth.

In the animation at the right, Venus actually completes two sidereal
periods (225 d) before it finally catches up with the Earth after
the synodic period (584 d).

For an outer planet, the sidereal period is (usually)longer
than the synodic period, because when the Earth returns to its
original position (one year) the planet has only moved slightly
in its orbit, and the Earth doesn't have to travel very
far to catch up to the planet.

Mars is an exception to this because it is so close to the Earth;
after one year it has already traveled more than half an orbit,
so the Earth has to complete two orbits before it can finally
catch up to Mars.

The telescope was invented by a Dutch optician late in the 16th century.

Galileo
Galilei (1561-1642), a professor of mathematics at the University
of Padua, heard about the telescope in 1609.

Recognizing the telescope's possibilities, Galileo immediately built one of his own, based only the sketchy details he had heard.

Galileo then improved the design of the telescope to the point where it could be used for astronomy.

Galileo quickly made several important astronomical discoveries,
which were published in 1610 in his book The Starry Messenger.

One of Galileo's observations was that Venus exhibited phases
similar to the Moon's:

Galileo noticed that Venus' phases were related to its angular
diameter and elongation: it is
smaller (farther away from us) at the gibbous phase and larger
(closer to us) at the crescent phase, with the extremes occurring
at small elongations.

Galileo also saw
the four large moons of Jupiter, now called the Galilean satellites.

The Galilean satellites were obviously orbiting Jupiter, which
was contrary to a basic assumption of the geocentric model, viz.
everything in the heavens orbited the Earth.

In 1616, when Copernicus' book was banned, Galileo was instructed
by the Vatican that he could only discuss the heliocentric model
as a "mathematical supposition" because anything else
would "restrict God's omnipotence".

Nevertheless, in 1632 Galileo published Dialogue Concerning
the Two Chief World Systems--Ptolemaic and Copernican, which
was such a masterpiece of exposition of the heliocentric model
that readers ignored the ordained conclusion.

Galileo was then brought before the Inquisition and forced to
publicly recant; his Dialogue was banned, and he spent
the last eight years of his life under house arrest.

The ban on Galileo's Dialogue wasn't lifted until 1822,
and the Vatican's censure of Galileo himself wasn't removed until
1992!

5.7 Kepler's Laws

(Discovering the Universe, 5th ed., §2-3)

Although the heliocentric model worked just as well as the
geocentric model, to make it work over a millenium Copernicus
still had to add epicycles.

The German astronomer Johannes
Kepler (1571-1630) had a different idea, however.

Kepler didn't believe planetary orbits were necessarily circles,
but could instead be other closed curves, such as the ellipse
or oval.

Recall that a circle is defined as the set of all
points that are a constant distance r (the radius)
from the centerC.

An ellipse is a generalization of a circle, involving
two points F1 and F2(each called
a focus, and together the foci) and two distances
r1
and r2,
whose sum is a constant:

r1
+ r2
=2a

When the foci coincide (coming together at the center), the result
is a circle with a radius a.

The constant 2a is equal to the length of the longer
or "major" axis, so a is called the semimajor
axis.

The semimajor axis therefore describes the overall size of the
ellipse.

It can be shown that a is the average distance of the
ellipse from one focus.

The constant c describes how far each focus is from the center, which determines how elongated the ellipse is (for a given value of a).
However, it is more useful to use the eccentricity:

e = c/a

Because c is always less than a, the value
of e varies between 0 and 1.

When e = 0, c = 0, the foci coincide, and we have
a circle.
When e =1, the foci approach the opposite ends of the
ellipse; the result is so elongated that, from one focus, both
the center and the other focus are infinitely far away, forming
a curve called a parabola.

Kepler
came to work with Tycho in 1600, and the latter's astronomical
records provided Kepler with the data he needed to test his hypothesis.
After many years of laborious calculations, Kepler was able to
demonstrate what is now known as Kepler's First Law:

Planetary orbits are ellipses, with the Sun at one focus.

Because the Sun is off-center, we can describe two special
positions on a planet's orbit, both on the major axis:
The perihelion is the point of closest approach to the
Sun; it is a distance a(1 - e) from the Sun.
The aphelion is the point where the planet is farthest
from the Sun; it is a distance a(1 + e) from the
Sun.

As can be seen in the table at the right, the eccentricity
of the planets' orbits is generally quite small, except for Mercury
and Pluto, whose orbits are noticeably elongated.

This is why circles initially worked well in describing planetary
orbits.

The table also shows the orbital inclination, or tilt,
of the planets' orbits relative to the ecliptic plane.

Planet

Semimajor
Axis a

Sidereal
Period P

Eccen-
tricity e

Orbital
Inclination

Mercury

0.3871 AU

0.2408 y

0.206

7.00°

Venus

0.7233 AU

0.6152 y

0.007

3.39°

Earth

1.0000 AU

1.0000 y

0.017

0.00°

Mars

1.5237 AU

1.8809 y

0.093

1.85°

Jupiter

5.2028 AU

11.862 y

0.048

1.31°

Saturn

9.5388 AU

29.458 y

0.056

2.49°

Uranus

19.1914 AU

84.01 y

0.046

0.77°

Neptune

30.0611 AU

164.79 y

0.010

1.77°

Pluto

39.5294 AU

248.5 y

0.248

17.15°

The orbital inclination is usually quite small, except for
Pluto.

Question: where did we
see orbital inclination previously?

Question: why doesn't
a planet usually disappear behind the Sun when they are in conjunction?

Kepler also noticed
another characteristic of planetary motion: planets move fastest
at perihelion, and slowest at aphelion.
Kepler was able to quantify these varying speeds in what is known
as Kepler's Second Law:

Planets sweep out equal areas in equal times.

Kepler published his First and Second Laws in 1609 in a book
entitled New Astronomy.

Ten years later, in 1619, Kepler discovered and published
an additional relationship.

Kepler's Third Law quantifies the observation that more
distant orbits have longer periods:

a3
= P2

Here, the semimajor axis a is measured in A.U. and the
orbital period P is measured in years.

The graph at the right shows log P vs. log a; the
data falls along a straight line, with a slope of 3/2.

Kepler also noticed that the Galilean satellites obeyed the
Third Law, as can be seen by the same 3/2 slope in the graph
at the right.

This implied that Kepler's Third Law was a general principle.

Galileo himself refused to accept Kepler's ideas, clinging
to the notion that planetary orbits must be circular, though
his reasons were based on his studies of motion rather than on
tradition.

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